A Banach algebra with its applications over paths of bounded variation.

*(English)*Zbl 1408.46046Following the study initiated by the author [J. Funct. Spaces 2018, Article ID 9345126, 10 p. (2018; Zbl 1400.46042)] where he introduced certain Banach algebras which generalized Cameron-Storvick’s Banach algebra, he now considers two new Banach algebras. He starts with two absolutely continuous real-valued functions \(\alpha\) and \(\beta\) defined on the interval \([0,T]\), where \(\beta\) is strictly increasing and \(|\alpha|'(t)+\beta'(t) >0\) for all \(t\in [0,T]\) and denotes by \(\nu_{\alpha,\beta}\) the Lebesgue-Stieltjes measure associated to \(|\alpha|+\beta\) and by \(L^2_{\alpha,\beta}([0,T])\) the Hilbert space of square integrable functions with respect to \(\nu_{\alpha,\beta}\).

Following K. S. Ryu [Honam Math. J. 32, No. 4, 633–642 (2010; Zbl 1209.28022)], the author recalls the construction of the analogue of a generalized Wiener measure according to a positive measure \(\phi\) on the Borel sets of \(\mathbb R\) which is denoted by \(w_{\alpha,\beta,\phi}\) and he considers the class \(\mathcal M(B([0,T])\) of measures of finite variation defined on subsets of \(B([0,T])\) (real right-continuous functions of bounded variation on \([0,T]\) that vanish at \(T\)) whose class of measurable sets is given by \(\{ v\in B([0,T]): \langle v, f\rangle_{\alpha,\beta}<\lambda\}\) for \(f\in L^2_{\alpha,\beta}([0,T])\) and \(\lambda\in \mathbb R\). It is known (see [R. H. Cameron and D. A. Storvick, Lect. Notes Math. 798, 18–67 (1980; Zbl 0439.28007)]) that \(\mathcal M(B([0,T])\) is a Banach algebra under convolution. He establishes the equivalence classes \(\mu_1\approx \mu_2\) if \(\int_{B([0,T])} J(x,v)\,d\mu_1(v)=\int_{B([0,T])} J(x,v)\,d\mu_2(v)\) for \(w_{\alpha,\beta,\phi}\)-almost everywhere \(x\in C([0,T])\), where \(J(x,v)=\exp\{i \int_0^T v(t)\,dx(t)\}\).

Denoting by \(\tilde{\mathcal M}(B([0,T])\) the set of equivalence classes, it is shown in the paper that \((\tilde{\mathcal M}(B([0,T]), \|\cdot\|)\) where \(\|[\mu]\|=\inf \{ \|\mu_1\|: \mu_1\approx \mu\}\), is a Banach algebra with unit. Also, the author defines the space of functions \(F(x)=\int_{B([0,T]} J(x,f)\,d\mu(f)\) for \(w_{\alpha,\beta,\phi}\)-almost everywhere \(x\in C([0,T])\) for some \(\mu\in \mathcal M(B([0,T])\). Defining \(\|F\|=\inf \{\|\mu\|\}\) where the infimum is taken over all possible \(\mu\), it is shown that such a space becomes a Banach algebra under pointwise multiplication isometrically isomorphic to \(\tilde{\mathcal M}(B([0,T])\).

Following K. S. Ryu [Honam Math. J. 32, No. 4, 633–642 (2010; Zbl 1209.28022)], the author recalls the construction of the analogue of a generalized Wiener measure according to a positive measure \(\phi\) on the Borel sets of \(\mathbb R\) which is denoted by \(w_{\alpha,\beta,\phi}\) and he considers the class \(\mathcal M(B([0,T])\) of measures of finite variation defined on subsets of \(B([0,T])\) (real right-continuous functions of bounded variation on \([0,T]\) that vanish at \(T\)) whose class of measurable sets is given by \(\{ v\in B([0,T]): \langle v, f\rangle_{\alpha,\beta}<\lambda\}\) for \(f\in L^2_{\alpha,\beta}([0,T])\) and \(\lambda\in \mathbb R\). It is known (see [R. H. Cameron and D. A. Storvick, Lect. Notes Math. 798, 18–67 (1980; Zbl 0439.28007)]) that \(\mathcal M(B([0,T])\) is a Banach algebra under convolution. He establishes the equivalence classes \(\mu_1\approx \mu_2\) if \(\int_{B([0,T])} J(x,v)\,d\mu_1(v)=\int_{B([0,T])} J(x,v)\,d\mu_2(v)\) for \(w_{\alpha,\beta,\phi}\)-almost everywhere \(x\in C([0,T])\), where \(J(x,v)=\exp\{i \int_0^T v(t)\,dx(t)\}\).

Denoting by \(\tilde{\mathcal M}(B([0,T])\) the set of equivalence classes, it is shown in the paper that \((\tilde{\mathcal M}(B([0,T]), \|\cdot\|)\) where \(\|[\mu]\|=\inf \{ \|\mu_1\|: \mu_1\approx \mu\}\), is a Banach algebra with unit. Also, the author defines the space of functions \(F(x)=\int_{B([0,T]} J(x,f)\,d\mu(f)\) for \(w_{\alpha,\beta,\phi}\)-almost everywhere \(x\in C([0,T])\) for some \(\mu\in \mathcal M(B([0,T])\). Defining \(\|F\|=\inf \{\|\mu\|\}\) where the infimum is taken over all possible \(\mu\), it is shown that such a space becomes a Banach algebra under pointwise multiplication isometrically isomorphic to \(\tilde{\mathcal M}(B([0,T])\).

Reviewer: Oscar Blasco (Valencia)

##### MSC:

46J10 | Banach algebras of continuous functions, function algebras |

28C20 | Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.) |

60H05 | Stochastic integrals |

46G12 | Measures and integration on abstract linear spaces |

##### Keywords:

Banach algebra; Feynman integral; Itô integral; Paley-Wiener-Zygmund integral; Wiener space**OpenURL**

##### References:

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